crazy dice

It is a standard exercise in elementary combinatorics to the number of ways of rolling any given value with 2 fair 6-sided dice (by taking the of the two rolls). The below table the number of such ways of rolling a given value n:

n

# of ways

2

1

3

2

4

3

5

4

6

5

7

6

8

5

9

4

10

3

11

2

12

1

A somewhat (un?)natural question is to ask whether or not there are any other ways of re-labeling the faces of the dice with positive integers that these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a re-labeling, via the labeling

Die ⁢1

={1,2,2,3,3,4}

Die ⁢2

={1,3,4,5,6,8}

and a pair of dice with this labeling are called a set of crazy dice. It is straight-forward to verify that the various possible occur with the same frequencies as given by the above table.